Wikipedia mentions that the word self in the word "self-inductance" is to differentiate it from "mutual inductance". But it does not state whether the two things are the same thing. So do the both terms mean that when I have a change in current in some thing such as coil or wire, there will be a voltage generated according to:
$V = L \frac{dI}{dt}$
where $L$ measures the inductance of the material.
"Mutual inductance" and "self-inductance" are not the same thing, no.
Suppose you have two loops (of wire, for example), loop A and loop B. When you run a current through loop A, it produces a magnetic field that fills the space around both loops. The mutual inductance between the two loops, denoted $M_{AB}$, is defined as the ratio of the magnetic flux through loop B to the current running around loop A:
$$M_{AB} = \frac{\Phi_B}{I_A}$$
You can do this calculation with loop A and loop B being two different loops, or you can do it using the same loop as both A and B (i.e. calculate the flux through the same loop that the current runs around). If you use two different loops, it's called the mutual inductance and labeled $M$ (as above); if you use just one loop, it's called the self-inductance and labeled $L$. "Inductance" is technically a general term that encompasses both possibilities, although it's common to use the word to mean self-inductance because that is the usual meaning.
